Documenta Math. 311

Arakelov Invariants of Riemann Surfaces

Robin de Jong

Received: August 25, 2004 Revised: June 16, 2005

Communicated by Thomas Peternell

Abstract. We derive closed formulas for the Arakelov-Green func- tion and the Faltings delta-invariant of a compact .

2000 Mathematics Subject Classification: 14G40, 14H42, 14H55. Keywords and Phrases: Arakelov theory, delta-invariant, Green func- tion, Weierstrass points.

1. Introduction The main goal of this paper is to give closed formulas for the Arakelov-Green function G and the Faltings delta-invariant δ of a compact Riemann surface. Both G and δ are of fundamental importance in the Arakelov theory of arith- metic surfaces [2] [8] and it is a central problem in this theory to relate these difficult invariants to more accessible ones. For example, in [8] Faltings gives formulas which relate G and δ for elliptic curves directly to theta functions and to the discriminant . Formulas of a similar explicit nature were derived by Bost in [3] for Riemann surfaces of genus 2. As to the case of general genus, less specific but still quite explicit formulas are known due to Bost [3] (for the Arakelov-Green function) and to Bost and Gillet-Soul´e[4] [10] (for the delta-invariant). We recall these results in Sections 2 and 4 below. In the present paper we express G and δ in terms of two new invariants S and T . Both S and T are initially defined as the norms of certain isomorphisms between line bundles, but eventually we find that they admit a very explicit description in terms of theta functions. They are intimately related to the divisor W of Weierstrass points. Of these new invariants, the T is certainly the easiest one. We are able to calculate it for hyperelliptic Riemann surfaces [13], where it is essentially a power of the Petersson norm of the discriminant modular form. The invariant S is less easy and involves a certain integral over the Riemann surface. We believe that the approach using S and T is very suitable for obtaining numerical results. An example at the end of this paper, where we compute δ and a special value of G for a certain hyperelliptic Riemann surface of genus 3, is meant to illustrate this belief.

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We start our discussion by recalling the definitions of G and δ. From now on until the end of section 4, we fix a compact Riemann surface X. Let g be its genus, which we assume to be positive. The space of holomorphic differentials 0 1 i H (X, ΩX ) carries a natural hermitian inner product (ω,η) 7→ 2 X ω ∧ η. We fix this inner product once and for all. Let {ω , . . . , ω } be an orthonormal basis 1 g R with respect to this inner product. We have then a fundamental (1,1)-form µ i g on X given by µ = 2g k=1 ωk ∧ ωk. It is verified immediately that the form µ does not depend on the choice of orthonormal basis, and hence is canonical. P Using this form, one defines the Arakelov-Green function G on X × X. This function gives the local intersections “at infinity” of two divisors in Arakelov theory [2].

Theorem 1.1. (Arakelov) There exists a unique function G : X × X → R≥0 satisfying the following properties for all P ∈ X: (i) the function log G(P, Q) is C∞ for Q 6= P ; (ii) we can write log G(P, Q) = log |zP (Q)| + f(Q) locally about P , where ∞ zP is a local coordinate about P and where f is C about P ; 2 (iii) we have ∂Q∂Q log G(P, Q) = 2πiµ(Q) for Q 6= P ;

(iv) we have X log G(P, Q)µ(Q) = 0. Theorem 1.1 isR proved in [2]. We call the function G the Arakelov-Green function of X. We note that by an application of Stokes’ theorem one can prove the symmetry relation G(P, Q) = G(Q, P ) for any P, Q ∈ X. Importantly, the Arakelov-Green function gives rise to certain canonical met- rics on line bundles on X. First, consider line bundles of the form OX (P ) with P a point on X. Let s be the canonical generating section of OX (P ).

We then define a smooth hermitian metric k · kOX (P ) on OX (P ) by putting kskOX (P )(Q) = G(P, Q) for any Q ∈ X. By property (iii) of the Arakelov- Green function, the curvature form of OX (P ) is equal to µ. Second, it is clear that the function G can be used to put a hermitian metric on the line bundle OX×X (∆X ), where ∆X is the diagonal on X × X, by putting ksk(P, Q) = G(P, Q) for the canonical generating section s of OX×X (∆X ). Restricting to the diagonal, we have a canonical adjunction isomorphism ∼ 1 1 OX×X (−∆X )|∆X −→ ΩX . We define a hermitian metric k · kAr on ΩX by insisting that this adjunction isomorphism be an isometry. It is proved in [2] 1 that this gives a smooth hermitian metric on ΩX , and that its curvature form is a multiple of µ. For the rest of the paper we shall take these metrics on OX (P ) 1 and ΩX (as well as on tensor product combinations of them) for granted and refer to them as Arakelov metrics. Next we introduce the Faltings delta-invariant. Let Hg be the generalised Siegel upper half plane of complex symmetric g × g-matrices with positive defi- nite imaginary part. Let τ ∈Hg be a period matrix associated to a symplectic g g g basis of H1(X, Z) and consider the analytic jacobian Jac(X) = C /Z + τZ associated to τ. We fix τ for the rest of our discussion. On Cg one has a t t theta function ϑ(z; τ) = n∈Zg exp(πi nτn + 2πi nz), giving rise to an ef- fective divisor Θ0 and a line bundle O(Θ0) on Jac(X). Now consider on the P

Documenta Mathematica 10 (2005) 311–329 Arakelov Invariants of Riemann Surfaces 313 other hand the set Picg−1(X) of divisor classes of degree g − 1 on X. It comes with a special subset Θ given by the classes of effective divisors. A fundamental theorem of Abel-Jacobi-Riemann says that there is a canonical ∼ bijection Picg−1(X) −→ Jac(X) mapping Θ onto Θ0. As a result, we can equip Picg−1(X) with the structure of a compact complex manifold, together with a divisor Θ and a line bundle O(Θ). We fix this structure for the rest of the discussion. The function ϑ is not well-defined on Picg−1(X) or Jac(X). We can remedy this by putting kϑk(z; τ) = (det Im τ)1/4 exp(−πty(Im τ)−1y)|ϑ(z; τ)|, with y = Im z. One can check that kϑk descends to a function on Jac(X). By our ∼ identification Picg−1(X) −→ Jac(X) we obtain kϑk as a function on Picg−1(X). It can be checked that this function is independent of the choice of τ. The delta-invariant is the constant appearing in the following theorem, due to Faltings (cf. [8], p. 402). Theorem 1.2. (Faltings) There is a constant δ = δ(X) depending only on X such that the following holds. Let {ω1, . . . , ωg} be an orthonormal basis of 0 1 H (X, ΩX ). Let P1,...,Pg, Q be points on X with P1,...,Pg pairwise distinct. Then the formula g k det ωk(Pl)kAr kϑk(P1 + · · · + Pg − Q) = exp(−δ(X)/8) · · G(Pk, Q) k

2. The Arakelov-Green function As was mentioned in the Introduction, the Weierstrass points of X play an important role in our approach to G and δ. The idea of considering Weierstrass points in the context of Arakelov theory is not new, cf. [6] and [14] for example. We start by recalling how we obtain the divisor of Weierstrass points using a Wronskian differential on X. Let {ψ1,...,ψg} be an (arbitrary) basis of

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0 1 H (X, ΩX ). Let P be a point on X and let z be a local coordinate about P . Write ψk = fk(z)dz for k = 1, . . . , g. We have a holomorphic function 1 dl−1f W (ψ) = det k z (l − 1)! dzl−1 µ ¶1≤k,l≤g locally about P from which we build the g(g+1)/2-fold holomorphic differential ⊗g(g+1)/2 ψ˜ = Wz(ψ)(dz) . We call ψ˜ the Wronskian differential about P and it is readily checked that ψ˜ is independent of the choice of the local coordinate. In fact, and this is less trivial, the differential ψ˜ extends over X to give a non-zero global section of the line ⊗g(g+1)/2 bundle ΩX . A change of basis changes the Wronskian differential by a non-zero scalar factor and hence the divisor of a Wronskian differential ψ˜ on X is unique. We denote this divisor by W, the divisor of Weierstrass points. This 3 divisor is effective and we have deg W = g − g. Writing W = P ∈X w(P ) · P we call the integer w(P ) the weight at P . This weight can be calculated using gap sequences, but we shall not need this. P Now fix for the moment a Q ∈ X. We consider the map φQ : X → Picg−1(X) given by sending P 7→ [gP − Q]. We put a smooth hermitian metric on O(Θ) by setting ksk = kϑk where s is the canonical generating section of O(Θ). We shall refer to this metric as the Arakelov metric on O(Θ). It can be verified ∗ by a short calculation using Riemann’s bilinear relations that φQO(Θ) is a line bundle on X of degree g3 and with curvature form a multiple of µ. In fact we can say more. It is a classical result (cf. for example [9], p. 31) that ∗ φQ(Θ) = W +g·Q. Hence we obtain the first statement of the next proposition. Proposition 2.1. We have a canonical isomorphism ∗ ∼ σQ : φQ(O(Θ)) −→ OX (W + g · Q) of line bundles on X. When both sides are equipped with their Arakelov metrics, the isomorphism σQ has constant norm on X. This norm is independent of the choice of Q. The proposition will be proven in the next section. Meanwhile, we observe that it leads quite quickly to a closed formula for G.

Definition 2.2. We define S(X) to be the norm of σQ for any Q ∈ X. In more concrete terms we have the following formula. Corollary 2.3. For any P, Q on X we have G(P, Q)g · G(P, W ) = S(X) · kϑk(gP − Q) , ∈W WY where the Weierstrass points are counted with their weights.

It follows from this corollary that the function W ∈W kϑk(gP − W ) does not vanish if P is not a . Hence the following formula makes sense. Q

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Theorem 2.4. For any P, Q on X with P not a Weierstrass point we have

g 1/g2 kϑk(gP − Q) G(P, Q) = S(X) · 1/g3 . W ∈W kϑk(gP − W ) Here the Weierstrass points are countedQ with their weights. Proof. This follows by applying the formula from Corollary 2.3 two times. First, take the (weighted) product over Q running through W. This gives 3 3 G(P, W )g = S(X)g −g · kϑk(gP − W ) . ∈W ∈W WY WY Plug this in again in the formula from Corollary 2.3. This gives 3 g −g 3 G(P, Q)g · S(X) g3 · kϑk(gP − W )1/g = S(X) · kϑk(gP − Q) , ∈W WY and a little rewriting gives the result. ¤ Taking logarithms in Corollary 2.3 and then integrating against µ with respect to the variable P immediately gives the following explicit formula for S(X). Theorem 2.5. For any fixed Q, the function log kϑk(gP − Q) is integrable against µ, and the formula

log S(X) = − log kϑk(gP − Q) · µ(P ) ZX holds. The invariant S(X) is readily calculated in the case g = 1. Example 2.6. Suppose that X = C/Z + τZ with Im τ > 0 is an . i The form µ is given by µ = 2 (dz ∧ dz)/Im τ and we have log S(X) = − log kϑk · µ . ZX A calculation (see [15], p. 45 or for a different approach [14], p. 250) yields log S(X) = − log((Im τ)1/4|η(τ)|) , 1/24 ∞ n where η(τ) is the usual Dedekind eta-function η(q) = q n=1(1 − q ) in q = exp(2πiτ). Q In the case that P = W is a Weierstrass point of X, the formula in Theorem 2.4 is still correct, provided that on the right hand side we take a limit for P approaching W . That this limit exists and that it indeed gives G(W, Q)g follows easily from the proof of Theorem 2.4. We finish this section by discussing very shortly several other approaches to G that we know from the literature. First of all, it is quite natural to develop G in terms of the eigenvalues and eigenfunctions of a Laplacian associated to µ on X. This is the approach taken in [8], see especially Section 3 of that paper. Second, it is possible to express G in terms of abelian differentials of the second and third kind, see for example [15], Chapter II. Third, and this is

Documenta Mathematica 10 (2005) 311–329 316 Robin de Jong perhaps most close to our approach since it also involves theta functions quite explicitly, there is an integral formula for G derived by Bost, cf. [3], Proposition 1. This interesting result reads as follows: let ν be the curvature form of O(Θ) on Picg−1(X). Then there is an invariant A(X) of X such that for every P, Q on X the formula 1 log G(P, Q) = log kϑk · νg−1 + A(X) g! ZΘ+P −Q holds. It would be interesting to have results that relate A(X) and S(X) to each other in a natural, conceptual way.

3. Proof of Proposition 2.1 Proposition 2.1 follows directly from Lemmas 3.1 and 3.2 below. We will be g 0 1 dealing, among other things, with the line bundle ∧ H (X, ΩX ) ⊗C OX on X. We equip this line bundle with the constant metric deriving from the hermitian i 0 1 inner product (ω,η) 7→ 2 X ω ∧ η on H (X, ΩX ) that we introduced in Section 1. From now on, this metric will be taken for granted and we shall also refer to it as an Arakelov metric.R Lemma 3.1. There is a canonical isomorphism of line bundles

⊗g(g+1)/2 g 0 1 ∨ ∼ ρ : ΩX ⊗ ∧ H (X, ΩX ) ⊗C OX −→ OX (W) on X. When both sides are equipped¡ with their Arakelov¢ metrics, the norm of this isomorphism is constant on X.

Proof. The Wronskian differential ψ˜ formed on an arbitrary basis {ψ1,...,ψg} 0 1 of H (X, ΩX ) leads to a morphism of line bundles

g 0 1 g(g+1)/2 ∧ H (X, ΩX ) ⊗C OX −→ ΩX by setting ξ1 ∧ . . . ∧ ξg ξ1 ∧ . . . ∧ ξg 7→ · ψ˜ . ψ1 ∧ . . . ∧ ψg

⊗g(g+1)/2 g 0 1 ∨ This gives a canonical section in ΩX ⊗ ∧ H (X, ΩX ) ⊗C OX whose divisor is W and thus we obtain the required isomorphism. The norm is con- ¡ ¢ stant on X because both sides have the same curvature form, and the divisors of their canonical sections are equal. ¤

Lemma 3.2. Let Q be an arbitrary point of X. There is a canonical isomor- phism of line bundles

∗ ∼ ⊗g(g+1)/2 g 0 1 ∨ φQ(O(Θ)) −→ ΩX ⊗ ∧ H (X, ΩX ) ⊗C OX ⊗ OX (g · Q) ³ ´ on X. When both sides are equipped¡ with their Arakelov¢ metrics, the norm of this isomorphism is constant on X and equal to exp(δ(X)/8).

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Proof. We are done once we prove that

g exp(δ(X)/8) · kϑk(gP − Q) = kω˜kAr(P ) · G(P, Q) for arbitrary P, Q on X, whereω ˜ is the Wronskian differential formed out of an 0 1 orthonormal basis {ω1, . . . , ωg} of H (X, ΩX ) and where the norm kω˜kAr ofω ˜ ⊗g(g+1)/2 is taken in the line bundle ΩX equipped with its Arakelov metric. The required formula follows from the formula in Theorem 1.2, by a computation which is also performed in [14], p. 233 and which runs as follows. Let P be a point on X, and choose a local coordinate z about P . By definition of 1 the Arakelov metric on ΩX we have that limQ→P |z(Q) − z(P )|/G(Q, P ) = kdzkAr(P ). Letting P1,...,Pg approach P in Theorem 1.2 we obtain

k det ω (P )k k det ω (P )k |z(Pk) − z(Pl)| lim k l Ar = lim k l Ar · k

= kω˜kAr(P ) . The required formula is therefore just a limiting case of Theorem 1.2 where all Pk approach P . ¤

4. The Faltings delta-invariant In the present section we will express the Faltings delta-invariant δ(X) in terms of S(X) and a second invariant T (X). The significance of our formula is that the constant T (X) is in a sense “classical” and easy to calculate numerically. To start our discussion, we observe that it follows from the previous sections that (multiples of) the divisor W of Weierstrass points appear as a divisor of a section of a line bundle in various different situations. We will take advantage of this fact and take combinations until we obtain an isomorphism of line bundles whose norm is easy to measure. First of all, recall (this is Proposition 2.1) that we have for any Q on X a canonical isomorphism

∗ ∼ σQ : φQ(O(Θ)) −→ OX (W + g · Q) . Taking the (weighted) tensor product over the Weierstrass points of X, we obtain a canonical isomorphism

∗ ∼ 3 σW : φW (O(Θ)) −→ OX (g · W) . ∈W WO Second, recall that by Lemma 3.1 we have a canonical isomorphism

⊗g(g+1)/2 g 0 1 ∨ ∼ ρ : ΩX ⊗ ∧ H (X, ΩX ) ⊗C OX −→ OX (W) . ¡ ¢ Documenta Mathematica 10 (2005) 311–329 318 Robin de Jong

Thirdly, taking a closer look at Lemma 3.2 we see that the proof in fact implies that we have on X × X a canonical isomorphism ∗ ∼ σ : Φ (O(Θ)) −→ OX×X (W · X + g · ∆X ) where Φ : X × X → Picg−1(X) is the map sending (P, Q) 7→ [gP − Q] and where again ∆X is the diagonal on X × X. Restricting σ to the diagonal, and using the adjunction isomorphism, we obtain a canonical isomorphism ∗ ⊗g ∼ σ|∆ : Φ (O(Θ))|∆X ⊗ ΩX −→ OX (W) .

Taking suitable combinations of σW , ρ and σ|∆ we obtain Proposition 4.1. There is a canonical isomorphism of (fractional) line bun- dles ∗ ⊗g ⊗(g+1) ∼ τ : Φ (O(Θ))|∆X ⊗ ΩX −→ ⊗(g−1)/g3 ¡ ¢ ⊗2 ∗ ⊗g(g+1)/2 g 0 1 ∨ φW (O(Θ)) ⊗ ΩX ⊗ ∧ H (X, ΩX ) ⊗C OX Ã W ! O ³ ¡ ¢ ´ on X. Our results thus far imply that τ has a constant norm for the Arakelov metrics on both sides. Definition 4.2. We define T (X) to be the norm of τ on X. The constant T (X) admits the following concrete description using a local coordinate. Proposition 4.3. Let P ∈ X not a Weierstrass point and let z be a local coordinate about P . Define kFzk(P ) as kϑk(gP − Q) kFzk(P ) = lim . Q→P |z(P ) − z(Q)|g

This limit exists and is non-zero. Further, let {ω1, . . . , ωg} be an orthonormal 0 1 basis of H (X, ΩX ). Then the formula −(g+1) (g−1)/g3 2 T (X) = kFzk(P ) · kϑk(gP − W ) · |Wz(ω)(P )| ∈W WY holds, where Wz(ω) is the determinant of the Wronskian of {ω1, . . . , ωg} with respect to z. In particular, the evaluation of T (X) for a given X only involves the evaluation of certain classical functions at an arbitrary (non-Weierstrass) point of X.

∗ ⊗g Proof. Let F be the canonical section of Φ (O(Θ))|∆X ⊗ ΩX coming from the canonical section in Φ∗(O(Θ)) and the canonical generating section of OX×X (∆X ) using the adjunction isomorphism. For its norm we have kF k = g kFzk · kdzkAr in the local coordinate z. We see from the isomorphism σ|∆

Documenta Mathematica 10 (2005) 311–329 Arakelov Invariants of Riemann Surfaces 319 that kF k(P ) does not vanish if P is not a Weierstrass point. Next, the canon- ∗ ical section of W ∈W φW O(Θ) has norm W ∈W kϑk(gP − W ) at P . Fi- ⊗g(g+1)/2 g 0 1 ∨ nally, the canonicalN section of ΩX ⊗ ∧Q H (X, ΩX ) ⊗C OX has norm g(g+1)/2 kω˜kAr = |Wz(ω)| · kdzkAr . The proposition¡ follows by taking¢ the appro- priate combinations of these norms. ¤

Considering the norms of the three isomorphisms σW , ρ and σ|∆ one sees that they are directly expressible in terms of exp(δ) and S(X). Hence the same holds for the norm T (X) of τ. Viewing things a little differently, we obtain a formula for exp(δ) in terms of S(X) and T (X). Theorem 4.4. The formula 3 exp(δ(X)/4) = S(X)−(g−1)/g · T (X) holds. g3−g Proof. The norm of σW is equal to S(X) . The norm of ρ is equal to S(X) exp(−δ(X)/8) as becomes clear by decomposing again the isomorphism from Proposition 2.1, which has norm S(X), into the isomorphisms from Lem- mas 3.1 and 3.2. Lastly, the norm of σ|∆ is equal to S(X) since σ has this norm and the restriction to the diagonal using the adjunction isomorphism is an isometry. We obtain the required formula by just combining. ¤ We want to finish this section with a second formula for T (X), involving only first order derivatives of the theta function. It is based on a function kJk on SymgX introduced by Gu`ardia in [11]. Let τ ∈Hg be a period matrix associated to a symplectic basis of H1(X, Z) and g g g consider again the analytic jacobian Jac(X) = C /Z + τZ . For w1,...,wg ∈ Cg we put ∂ϑ J(w ,...,w ) = det (w ) 1 g ∂z l µ k ¶ and g g+2 t −1 kJk(w1,...,wg) = (det Im τ) 4 exp(−π yk(Im τ) yk) · |J(w1,...,wg)| . kX=1 Here in the latter formula yk = Im wk for k = 1, . . . , g. It can be checked that the function kJk(w1,...,wg) depends only on the classes in Jac(X) of the vectors wk. Now let P1,...,Pg be a set of g points on X. We take vectors g w1,...,wg ∈ C such that for all k = 1, . . . , g the class [wk] ∈ Jac(X) corre- g sponds to [ l=1 Pl] ∈ Picg−1(X) under the Abel-Jacobi-Riemann correspon- l6=k dence Picg−P1(X) ↔ Jac(X). We then put kJk(P1,...,Pg) = kJk(w1,...,wg). One may check that this does not depend on the choice of the period ma- trix τ. The function kJk has the following geometrical property: we have kJk(P1,...,Pg) = 0 if and only if P1,...,Pg are linearly dependent on the 0 1 ∨ image of X under the canonical map X → P(H (X, ΩX ) ). We refer to [11] for a proof of the following theorem.

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Theorem 4.5. Let P1,...,Pg, Q be points on X with P1,...,Pg pairwise dis- tinct. Then the formula g G(P , Q)g−1 kϑk(P + · · · + P − Q)g−1 = exp(δ(X)/8) · kJk(P ,...,P ) · k=1 k 1 g 1 g G(P ,P ) Q k

Proposition 4.6. Let P1,...,Pg, Q be generic points on X. Then the formula kϑk(P + · · · + P − Q) 2g−2 T (X) = 1 g · g kϑk(gP − Q)1/g µ k=1 k ¶ 1/g g Q k=6 l kϑk(gPk − Pl) (g−1)/g4 · 2 · kϑk(gPk − W ) kJk(P1,...,Pg) ÃQ ! W ∈W k=1 Y Y holds. Again the Weierstrass points are counted with their weights. Let us make the invariant T (X) explicit in the case that X is an elliptic curve. Writing X = C/Z + τZ with Im τ > 0 we obtain from either Proposition 4.3 or 4.6 that ∂ϑ 1 + τ T (X) = (Im τ)−3/2 exp(πIm τ/2) · | ( ; τ)|−2 . ∂z 2 By Jacobi’s derivative formula (cf. [19], Chapter I, §13) we can rewrite this as T (X) = (2π)−2 · ((Im τ)6|∆(τ)|)−1/4 24 ∞ n 24 where ∆ is the discriminant modular form ∆(q) = η(q) = q n=1(1 − q ) in q = exp(2πiτ). Using Theorem 4.4 we obtain Q δ(X) = − log((Im τ)6|∆(τ)|) − 8 log(2π) which is well-known, see [8], p. 417. In [13] we obtain a generalisation of the above formula for T (X) to the case where X is a hyperelliptic Riemann surface of genus g ≥ 2. The result is expressed in terms of the discriminant modular form ϕg on the generalised Siegel upper half plane Hg as defined in [17], Section 3. This is a modular form 2g+1 on Γg(2) = {γ ∈ Sp(2g, Z) : γ ≡ I2g mod 2} of weight 4r, where r = g+1 , generalising the usual discriminant modular form ∆ in genus 1. ¡ ¢ Theorem 4.7. Let X be a hyperelliptic Riemann surface of genus g ≥ 2. Choose an ordering of the Weierstrass points on X and construct a canonical symplectic basis of H1(X, Z) starting with this ordering (cf. [19], Chapter IIIa, §5). Let τ ∈Hg be a period matrix of X associated to this canonical basis and −(4g+4)n 2g put ∆g(τ) = 2 · ϕg(τ) where n = g+1 . Then ∆g(τ) is non-zero and the formula ¡ ¢ 3g−1 −2g 2r − T (X) = (2π) · ((Im τ) |∆g(τ)|) 8ng holds.

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It is an intriguing question whether the invariant T (X) admits of a simple description in terms of modular forms for a general Riemann surface X of genus g. We finish this section by remarking that a closed formula of a quite different type can be given for δ using work of Bost [4] and Gillet-Soul´e[10]. The point of view leading to this formula is the Riemannian manifold structure on X deriving from µ. Let ds2 be the metric on X given in conformal coordinates by ds2 = −2 ′ 2hzzdzdz with hzz = kdzkAr . Let det ∆h be the zeta regularised determinant of the Laplace operator with respect to this metric, and let vol(X,h) be the volume of X. Then the formula det ∆′ δ(X) = c(g) − 6 log h vol(X,h) holds, where c(g) is a constant depending only on g. It would be interesting to know whether the terms occurring in this formula can be naturally related to the constants S(X) and T (X) which are the subject of this paper.

5. Applications to intersection theory In this section we discuss several applications of our results to Arakelov inter- section theory. Let p : X → B be an over the spectrum B of the ring of integers of a number field K. For us this means that X is a regular and that p is a proper and flat relative curve whose generic fiber is smooth and geometrically connected. We denote this generic fiber by XK . We assume that the reader is familiar with the basic notions and statements in the Arakelov intersection theory on X , as explained in [2] or [8]. We let g be the genus of XK , and assume that it is positive. We fix a K- basis {ψ1,...,ψg} of regular differential 1-forms on XK . Looking back at the discussion at the beginning of Section 2, which was purely algebraic, we note that a non-zero Wronskian differential ψ˜ can be formed out of this basis. Its divisor div ψ˜ is an effective K-divisor on XK and we have, completely analogous to Lemma 3.1, a canonical isomorphism ∨ ∼ Ω⊗g(g+1)/2 ⊗ ∧gH0(X , Ω1 ) ⊗ O −→ O (div ψ˜) XK OXK K XK K XK XK of invertible sheaves on X¡K . We denote by W the¢ Zariski closure of div ψ˜ in X . Let ωX /B be the relative dualising sheaf of p : X → B. Lemma 5.1. The above isomorphism extends to a canonical isomorphism

⊗g(g+1)/2 ∗ ∨ ∼ ρ : ωX /B ⊗OX p (det p∗ωX /B) −→ OX (V + W) of invertible sheaves on X , for some¡ effective divisor¢ V whose support is entirely contained in the fibers of p. Proof. The idea for the proof is taken from [1], p. 1298, where an analogous result is proven for the function field case. We recall that ψ˜ is given in a local

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⊗g(g+1)/2 coordinate z by Wz(ψ)(dz) where 1 dl−1f W (ψ) = det k z (l − 1)! dzl−1 µ ¶1≤k,l≤g if the ψk are locally written as ψk = fk(z)dz for k = 1, . . . , g. On XK this gives rise to a morphism of invertible sheaves g 0 1 ⊗g(g+1)/2 ∧ H (XK , ΩXK ) ⊗K OXK −→ ΩXK by setting ξ1 ∧ . . . ∧ ξg ξ1 ∧ . . . ∧ ξg 7→ · ψ˜ ψ1 ∧ . . . ∧ ψg (cf. the proof of Lemma 3.1). Now note that the construction of ψ˜ is valid for smooth proper curves over any base scheme. As a result, by modifying the basis {ψ1,...,ψg} if necessary, the above morphism extends canonically at least over the open dense subscheme of X where p is smooth. Automatically it extends ∗ ⊗g(g+1)/2 then further to give a canonical morphism p (det p∗ωX /B) → ωX /B on the ∗ ∨ whole of X . Multiplying by (p (det p∗ωX /B)) we obtain from this a morphism ⊗g(g+1)/2 ∗ ∨ OX −→ ωX /B ⊗OX p (det p∗ωX /B) . The image of 1 is a section whose divisor is¡ a divisor V +W¢ where V is effective and has support entirely contained in the fibers of p. This gives the lemma. ¤ The divisor V is an invariant of the arithmetic surface p : X → B and we shall use it without further mention in the sequel. ∗ Example 5.2. In the case that g = 1, the morphism p p∗ωX /B → ωX /B in the above proof is just the natural morphism, as is readily checked. According to [16], Corollary 3.27, if p : X → B is a minimal arithmetic surface, then the ∗ natural morphism p p∗ωX /B → ωX /B is in fact an isomorphism. Hence we find V = ∅ in this case. We want to translate the isomorphism ρ of Lemma 5.1 into an equality of Arakelov divisors on X . For this we need a notation for the norm of ρ at the various complex embeddings of K. Definition 5.3. Let X be a compact Riemann surface of positive genus. We denote by R(X) the norm of the isomorphism ρ from Lemma 3.1. It follows from our discussion so far that R(X) = S(X) · exp(−δ(X)/8). Now let’s turn back to our arithmetic surface p : X → B. We recall from [2] [8] that both sides of the isomorphism ρ from Lemma 5.1 come equipped with a canonical structure of metrised invertible sheaf, and that to each non-zero rational section of such a sheaf we can associate its Arakelov divisor. For each complex embedding σ of K we denote by Xσ the compact Riemann surface (XK ⊗K,σ C)(C) obtained from base changing XK to C along σ. We denote by Fσ the corresponding Arakelov fiber. The next proposition follows easily from Lemma 5.1 and from the fact that ρ has constant norm R(Xσ) on Xσ.

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Proposition 5.4. We have an equality 1 g(g + 1)ω = V + W + log R(X ) · F + p∗(det p ω ) 2 X /B σ σ ∗ X /B σ X of Arakelov divisors on X . Here the sum runs over the complex embeddings of K. This proposition can be used to deduce some interesting formulas involving Arakelov intersection numbers. Definition 5.5. We define a function R on the set of closed fibers of p : X → B as follows. Let s be a closed point of B. If g = 1, we put R(Xs) = 0. If g ≥ 2, then we define R(Xs) by the equality (2g−2)·log R(Xs) = (Vs, ωX /B)·log #k(s), where (Vs, ωX /B) is the usual intersection number of the divisors V and ωX /B above s, and where k(s) is the residue field at s. As the next proposition implies, the function R can be seen as an analogue of the previously defined R for compact Riemann surfaces. The quantity deg det p∗ωX /B is the usual Arakelov degree of the metrised invertible sheaf det p∗ωX /B on B (i.e., [K : Q] times the Faltings height of p : X → B). d Proposition 5.6. Assume that p : X → B is a semi-stable arithmetic surface. Then for the self-intersection of the relative dualising sheaf we have a lower bound 8(g − 1) (ω, ω) ≥ · log R(X ) + log R(X ) + deg det p ω . (2g − 1)(g + 1) s σ ∗ X /B Ã s σ ! X X Here the first sum runs over the closed points s ∈ B, and thed second sum runs over the complex embeddings of K. Proof. In the case g = 1, the lower bound is trivially satisfied since we have (ωX /B, ωX /B) = 0 in this case by [8], Theorem 7. So assume that g ≥ 2. We take the equality from Proposition 5.1 and intersect the divisors on both sides 1 with ωX /B. This gives that 2 g(g + 1)(ωX /B, ωX /B) can be written as

(W, ωX /B) + (2g − 2) · log R(Xs) + log R(Xσ) + deg det p∗ωX /B . Ã s σ ! X X For the term (W, ωX /B) we have by [8], Theorem 5 the lowerd bound g3 − g 1 (W, ω ) ≥ (ω , ω ) = (g + 1)(ω , ω ) X /B 2g(2g − 2) X /B X /B 4 X /B X /B since the generic degree of W is g3 − g. Using this in the first equality gives the required lower bound. ¤ We remark that for a semi-stable arithmetic surface p : X → B the num- bers log R(Xs) are always non-negative. Lower bounds of a similar type for (ωX /B, ωX /B) can be found in [6]. The problem with the above proposition is that the right hand side may be negative, and then the lower bound becomes

Documenta Mathematica 10 (2005) 311–329 324 Robin de Jong useless in view of the fundamental inequality (ωX /B, ωX /B) ≥ 0 proved by Faltings [8]. However, for any fixed g ≥ 2 the invariant log R(X) can become arbitrarily large, as the next proposition shows.

Proposition 5.7. Let Xt be a holomorphic family of compact Riemann sur- faces of genus g ≥ 2 over the punctured disk, degenerating as t → 0 to the union of two Riemann surfaces of positive genera g1, g2 with two points identified. Suppose that neither of these two points was a Weierstrass point. Then the formula g g log R(X ) = − 1 2 log |t| + O(1) t 2g holds as t → 0. For a proof we refer to the author’s thesis [12]. The next application we have in mind is a formula for the self-intersection of a point. In order to derive this formula it is convenient to use the machinery of the determinant of cohomology det Rp∗(·) and the Deligne bracket h·, ·i, for which we refer to [7]. We will use that for any section P : B →X and any in- ∼ ∗ vertible sheaf L on X we have canonical isomorphisms hOX (P ), Li −→ P L ∼ ⊗−1 and hP, ωX /Bi −→hP,P i . The latter is sometimes called the adjunc- tion formula. Moreover, we have a Riemann-Roch theorem in the follow- ing form: for each invertible sheaf L on X there is a canonical isomorphism ⊗2 ∼ −1 ⊗2 (det Rp∗L) −→hL, L ⊗ ωX /Bi ⊗ (det p∗ωX /B) . Lemma 5.8. Let P be a section of p, not a Weierstrass point on the generic fiber. Then we have a canonical isomorphism

∗ ⊗2 ∼ ⊗−2 υ : P (OX (V + W)) −→ (det Rp∗OX (gP )) of line bundles on B. When restricted to the generic fiber, the left hand ⊗2 side gets identified with OSpecK and the right hand side gets identified with 0 ⊗−2 H (XK , OXK (gP )) . The latter has a canonical trivialising section 1 and ⊗2 the isomorphism υ, when restricted to the generic fiber, sends the 1 of OSpecK 0 ⊗−2 to the 1 of H (XK , OXK (gP )) .

Proof. The Riemann-Roch theorem applied to the invertible sheaf OX (gP ) gives a canonical isomorphism

⊗2 ∼ −1 ⊗2 (det Rp∗OX (gP )) −→hOX (gP ), OX (gP ) ⊗ ωX /Bi ⊗ (det p∗ωX /B) . By the , the right hand side can be canonically identified ⊗g(g+1) ⊗2 with hP,P i ⊗ (det p∗ωX /B) , giving a canonical isomorphism

⊗2 ∼ ⊗g(g+1) ⊗2 (det Rp∗OX (gP )) −→hP,P i ⊗ (det p∗ωX /B) . On the other hand, pulling back the isomorphism ρ from Lemma 5.1 along P and using once more the adjunction formula we find a canonical isomorphism ⊗−g(g+1)/2 ∼ hP,P i −→hV + W,P i ⊗ det p∗ωX /B

Documenta Mathematica 10 (2005) 311–329 Arakelov Invariants of Riemann Surfaces 325 and hence ⊗g(g+1) ∼ ∗ ⊗−2 ⊗−2 hP,P i −→ (P OX (V + W)) ⊗ (det p∗ωX /B) . The isomorphism υ follows by combining these isomorphisms. Since P is 0 not a Weierstrass point on the generic fiber, we have that H (XK ,gP ) is 1-dimensional and hence is generated by its canonical section 1. The last statement of the lemma follows then by carefully spelling out all the isomor- phisms. ¤ Proposition 5.9. Let P be a section of p, not a Weierstrass point on the 1 generic fiber. Then R p∗OX (gP ) is a torsion module on B and the self- 1 intersection − 2 g(g + 1)(P,P ) is given by 1 − log G(Pσ, Wσ) + log #R p∗OX (gP ) + log R(Xσ) + deg det p∗ωX /B . σ σ X X Here σ runs through the complex embeddings of K. d

1 Proof. That R p∗OX (gP ) is a torsion module on B follows since we have 1 H (XK ,gP ) = 0 on the generic fiber. As to the formula, we take the equality from Proposition 5.1 and intersect the divisors on both sides with P . By the Arakelov adjunction formula (ω,P ) = −(P,P ) we obtain 1 − g(g + 1)(P,P ) = (V + W,P ) + log R(X ) + deg det p ω . 2 σ ∗ X /B σ X 1 It remains therefore to see that (V + W,P )fin = log #Rdp∗OX (gP ). For this we invoke Lemma 5.8. It follows from the description of υ on the generic fiber that in fact υ is the natural isomorphism over the open dense subscheme of B where P does not meet V + W. Now for any closed point s of B denote by es 1 the length at s of R p∗OX (gP ). Then if we let D = s es · s, the invertible 1 0 sheaf det R p∗OX (gP ) gets identified with OB(D) and, since det R p∗OX (gP ) P is trivialised by the section 1, the determinant of cohomology det Rp∗OX (gP ) gets identified with OB(−D). By Lemma 5.8, for any closed point s the length es coincides with the intersection multiplicity of P and V + W at s and conse- 1 ¤ quently (V + W,P )fin = s es log #k(s) = log #R p∗OX (gP ).

P6. A numerical example In this section we use the results of Sections 2 and 4 to calculate the Falt- ings height and the self-intersection of the relative dualising sheaf of a certain of genus 3 defined over the rationals. We start with two theoretical results, both of which can be proved by methods similar to those used in [5], Section 3. Let K be a number field, and let OK be its ring of integers. For a non- zero element a ∈ OK and a prime ideal ℘ of OK we denote by v℘(a) the exponent of ℘ in the prime ideal decomposition of a · OK . Let f ∈ OK [x] be a monic polynomial of degree 5 with f(0) and f(1) units in OK and put g(x) = x(x − 1) + 4f(x).

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Proposition 6.1. Suppose that the discriminant ∆ of g is non-zero, that we have v℘(∆) = 0 or 1 for each prime ideal ℘ of residue characteristic 6= 2, and that g mod ℘ has a unique multiple root of multiplicity 2 for prime ideals ℘ with residue characteristic 6= 2 and with v℘(∆) = 1. Then the equation 2 Cf : y = x(x − 1)f(x) defines a hyperelliptic curve of genus 3 over K. It extends to a semi-stable arithmetic surface p : X → B = Spec(OK ). We have that X has bad reduction at ℘ if and only if ℘ has residue characteristic 6= 2 and v℘(∆) = 1. For such ℘, the fiber at ℘ is an irreducible curve with a single double point. The differentials 2 dx/y, xdx/y, x dx/y form a basis of the free OB-module p∗ωX /B. The points W0, W1 on Cf given by x = 0 and x = 1 extend to disjoint sections of p. As to the Faltings height and the self-intersection of the relative dualising sheaf of Cf we have the following. For a complex embedding σ of K we denote by Xf,σ the compact Riemann surface (Cf ⊗K,σ C)(C) obtained by base changing Cf to C along σ. For each σ, we choose a symplectic basis of H1(Xf,σ, Z) and 2 form the period matrix Ωσ = (Ω1,σ|Ω2σ) for dx/y, xdx/y and x dx/y on this −1 basis. We further put τσ = Ω1σ Ω2σ.

Proposition 6.2. The degree of det p∗ωX /B satisfies 1 deg det p ω = − log | det Ω |2(det Im τ ) . ∗ X /B 2 1σ σ σ X ¡ ¢ For the self-intersectiond of the relative dualising sheaf we have the formula

(ωX /B, ωX /B) = 24 log Gσ(W0, W1) , σ X where Gσ denotes the Arakelov-Green function on Xf,σ. We apply these propositions to a concrete example. We choose K = Q and f(x) = x5 + 6x4 + 4x3 − 6x2 − 5x − 1. It can be checked that g(x) = x(x − 1) + 4f(x) satisfies the conditions of Proposition 6.1. The correspond- ing hyperelliptic curve Cf has bad reduction at the primes p = 37,p = 701 and p = 14717. Let Xf be the compact Riemann surface obtained from base changing Cf to the complex numbers. We choose an ordering of the Weier- strass points of X and as in [19], Chapter IIIa, §5 this gives us a canonical way to construct a symplectic basis for H1(Xf , Z). We have computed the periods with respect to this basis of the differentials dx/y, xdx/y and x2dx/y. Using Proposition 6.2 we easily obtain

deg det p∗ωX /B = −1.280295247656532068 . . . which is the Faltings height of Cf . Next we take a look at the self-intersection of the relative dualisingd sheaf. According to Proposition 6.2 we need to calculate G(W0, W1). We apply Theorem 2.4 where we carefully take a limit for P approaching W0. Using theory as developed for example in [19], Chapter IIIa it is possible to make the Abel-Jacobi-Riemann correspondence Pic2(Xf ) ↔

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Jac(Xf ) completely explicit. This makes it easy to carry out the theta function evaluations that are needed to compute G(W0, W1). The calculation of S(Xf ) is, however, considerably harder. We recall that in our case S(Xf ) is given by

log S(Xf ) = − log kϑk(3P − Q) · µ(P ) ZXf where µ is the Arakelov metric and where Q is any point on Xf . We want to express µ in terms of the coordinates x,y but then it immediately becomes clear that the integrand will diverge at the Weierstrass point at infinity. However, by taking logarithms in Theorem 2.4 and integrating against µ(Q) we find the alternative formula 1 log S(X ) = −9 log kϑk(3P − Q) · µ(Q) + log kϑk(3P − W ) , f 3 X ∈W Z WX valid for any non-Weierstrass point P on X, in which the integrand behaves better. In fact, the integrand now only has a singularity at Q = P . Let −1 Ω = (Ω1|Ω2) be the period matrix of Xf referred to above and put τ = Ω1 Ω2. t −1 Let (µkl) be the matrix given by (µkl) = Ω1(Im τ) Ω1 . An application of i Riemann’s bilinear relations yields µ = 6 µkl ψk ∧ ψl with ψ1 = dx/y, ψ2 = 2 ¡ ¢ xdx/y and ψ3 = x dx/y. Writing x = u + iv with u,v ∈ R we can rewrite this as the real 2-form P 1 µ = µ + 2µ u + 2µ (u2 − v2) + µ (u2 + v2) 3 11 12 13 22 ¡ dudv +2µ u(u2 + v2) + µ (u2 + v2)2 · , 23 33 |h(u + iv)| where h(x) = x(x − 1)g(x). Using a computer algebra¢ package, we have eval- uated the integral. This is a slow process, because one has to take care of the logarithmic singularity. On the other hand, it is possible to check the answers by trying various choices of P . We found that within reasonable time limits we can only reach an accuracy within ± 0.005. The end result is

log S(Xf ) = 17.57 . . . Using this we find the approximation

G(W0, W1) = 2.33 . . . and finally

(ωX /B, ωX /B) = 20.32 . . .

It is almost no extra effort to compute also the delta-invariant of Xf . Using Theorem 4.7 we obtain, first of all,

log T (Xf ) = −4.44361200473681284 . . .

With Theorem 4.4 and our value above for log S(Xf ) we get as a result

δ(Xf ) = −33.40 . . .

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The reader may check that the Noether formula [18] is verified by our numerical results.

Acknowledgments. The author wishes to thank Gerard van der Geer for his encouragement and helpful remarks.

References [1] S. Y. Arakelov, Families of algebraic curves with fixed degeneracies, Math. USSR Izvestija 5 (1971), 1277–1302. [2] S. Y. Arakelov, An intersection theory for divisors on an arithmetic surface, Math. USSR Izvestija 8 (1974), 1167–1180. [3] J.-B. Bost, Fonctions de Green-Arakelov, fonctions thˆeta et courbes de genre 2, C.R. Acad. Sci. Paris Ser. I 305 (1987), 643–646. [4] J.-B. Bost, Conformal and holomorphic anomalies on Riemann surfaces and determinant line bundles. In: M. Mebkhout, R. Seneor (eds.), 8th International Congress on Mathematical Physics, World Scientific 1987. [5] J.-B. Bost, J.-F. Mestre, L. Moret-Bailly, Sur le calcul explicite des “classes de Chern” des surfaces arithm´etiques de genre 2. In: S´eminaire sur les pinceaux de courbes elliptiques, Ast´erisque 183 (1990), 69–105. [6] J.-F. Burnol, Weierstrass points on arithmetic surfaces, Inv. Math. 107 (1992), 421–432. [7] P. Deligne, Le d´eterminant de la cohomologie. In: Contemporary Mathe- matics vol. 67, American Mathematical Society (1987), 93–177. [8] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119 (1984), 387–424. [9] J.D. Fay, Theta functions on Riemann surfaces. Lect. Notes in Math. vol. 352, Springer-Verlag 1973. [10] H. Gillet, C. Soul´e, An arithmetic Riemann-Roch theorem, Inv. Math. 110 (1992), 473–543. [11] J. Gu`ardia, Analytic invariants in Arakelov theory for curves, C.R. Acad. Sci. Paris Ser. I 329 (1999), 41–46. [12] R. de Jong, Explicit Arakelov geometry. Thesis University of Amsterdam, 2004. [13] R. de Jong, Faltings’ delta-invariant of a hyperelliptic Riemann surface. In: G. van der Geer, B. Moonen, R. Schoof (eds.), Proceedings of the Texel Conference “The analogy between number fields and function fields”, Birkh¨auser Verlag 2005. [14] J. Jorgenson, Asymptotic behavior of Faltings’s delta function, Duke Math. J. 61 (1990), 1, 303–328. [15] S. Lang, Introduction to Arakelov theory. Springer-Verlag 1988. [16] Q. Liu, and Arithmetic Curves. Oxford Graduate Texts in Mathematics 6, Oxford Science Publications 2002. [17] P. Lockhart, On the discriminant of a hyperelliptic curve, Trans. Amer. Math. Soc. 342 (1994), 2, 729–752.

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[18] L. Moret-Bailly, La formule de Noether pour les surfaces arithm´etiques, Inv. Math. 98 (1989), 491–498. [19] D. Mumford, Tata Lectures on Theta I,II. Progress in Mathematics vol. 28, 43, Birkh¨auser Verlag 1984. [20] R. Wentworth, The asymptotics of the Arakelov-Green’s function and Falt- ings’ delta invariant, Commun. Math. Phys. 137 (1991), 427–459.

Robin de Jong Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden The Netherlands [email protected]

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